A visualisation for conveying the dynamics of iterative eigenvalue algorithms over PSD matrices

TL;DR

This paper introduces a novel visualization method for the dynamics of iterative eigenvalue algorithms on PSD matrices, enhancing understanding of convergence and stability for students and researchers.

Contribution

It presents a new visualization approach for iterative eigenvalue algorithms, providing intuitive analysis and a proof of a general theorem in the field.

Findings

Fixed points are visually identified and analyzed.

Convergence speed depends on whether eigenvalues or eigenvectors are targeted.

A theorem on the dynamics of iterative eigenvalue algorithms is proved.

Abstract

We propose a new way of visualising the dynamics of iterative eigenvalue algorithms such as the QR algorithm, over the important special case of PSD (positive semi-definite) matrices. Many subtle and important properties of such algorithms are easily found this way. We believe that this may have pedagogical value to both students and researchers of numerical linear algebra. The fixed points of iterative algorithms are obtained visually, and their stability is analysed intuitively. It becomes clear that what it means for an iterative eigenvalue algorithm to "converge quickly" is an ambiguous question, depending on whether eigenvalues or eigenvectors are being sought. The presentation is likely a novel one, and using it, a theorem about the dynamics of general iterative eigenvalue algorithms is proved. There is an accompanying video series, currently hosted on Youtube, that has certain advantages in terms of fully exploiting the interactivity of the visualisation.